First. When Cersei and Ned talked in the godswood in aGoT,
she mentioned Jon, and wondered who his mother was, (paraphrasing) "...Some peasant wife you raped, while her holdfast burned?" This indicates that there were fightings in Dorne when Ned went there to get Lyanna back. But I thought the Martells stayed out of the war, and that Ned went there when the war was all over. So: did Ned take an army with him into Dorne, or not?

Ned's army did not accompany him to Dorne, no. There were no battles in Dorne during Robert's Rebellion, though doubtless there were minor skirmishes along the borders. But it's not entirely correct that the Martells stayed out of the war. Rhaegar had Dornish troops with him on the Trident, under the command of Prince Lewyn of the Kingsguard. However, the Dornishmen did not support him as strongly as they might have, in part because of anger at his treatment of Elia, in part because of Prince Doran's innate caution. Cersei's line reflects no more than a desire to wound, to say something nasty to get a rise out of Ned.

Second: We discussed whether Robert loved his brother Renly or not. Haaruk thought so, while I never envisioned their relationship as more than lukewarm. (Jaime said Robert hardly could stomach his brothers (plural form)). Which is correct?

There are many different kinds of love. Robert was dutiful toward his brothers, and no doubt loved them in a way... but he didn't necessarily like them. His relations with Stannis were always prickly. Renly was the baby of the family, and spent little time in Robert's company until he was old enough to come to court. I suspect Robert was fond of the boy, but not especially close to him.

Stannis always resented being given Dragonstone while Renly got Storm's End, and took that as a slight... but it's not necessarily true that Robert meant it that way. The Targaryen heir apparent had always been titled Prince of Dragonstone. By making Stannis the Lord of Dragonstone, Robert affirmed his brother's status as heir (which he was, until Joff's birth a few years later). Robert could just as lawfully retained both castles for his sons, and made Joffrey the Prince of Dragonstone and Tommen the Lord of Storm's End. Giving them to his brothers instead was another instance of his great, but rather careless, generosity.